Some say adding constraints cuts the feasible region smaller hence the same solver terminates faster due to the less search effort. Others say it adds more complexity to the problem and it may take longer to solve.
Personally, I think it depends on the model and the algorithm that is solving it. For instance, adding constraints to a linear program may add more vertices to the polyhedron that represents the feasible region. Consequently, the Simplex algorithm may take longer as it pivots over more extreme points.
To what extent do we know the effect of adding constraint to any given mathematical model and solution algorithm?